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If a helicopter hovers in a fixed position for 24 hours will the earth rotate around it?

Perhaps you're familiar with the idea of inertia: an object in motion tends to stay in motion (unless acted upon by a net external force). In a way, we can also refer to this idea as conservation of momentum. That idea's going to play a big part in answering your question. When the helicopter starts out, it's sitting on the ground and the ground (being part of the Earth) is rotating at one revolution per day, as we know. Since the helicopter is also sitting on the ground, it's also inside this moving reference frame, and has the momentum that goes with it so the helicopter is also moving at one revolution per day. In fact, so is the air! Now, when the helicopter takes off, it flies straight up to some height above the Earth's surface. But though the helicopter has exerted a force (through the use of its rotors) to lift it straight up, it hasn't exerted a force in the horizontal direction to counter the motion (momentum) it already had that one revolution per minute! So though the helicopter is no longer touching the ground, unless the pilot purposely exerts a force against the helicopter's initial momentum, the helicopter will continue to move at one revolution per day, and thus remain above the same spot on the Earth's surface from where it took off. The momentum that the helicopter started with is the same as what it ends with that's conservation of momentum! The same is true on a smaller scale when you jump in the air if you jump straight up, you'll land exactly where you started, because in every other direction (except up and down), your momentum is the same (try it out the next time you're on a plane: the plane acts like a miniature version of the Earth, and when you jump, you land right where you were, even though the plane's going 500 miles an hour!). On a larger scale, rocket scientists have to account for the motion of the Earth before they launch a satellite. In order to put the satellite into a specific orbit, they can't just shoot it straight up from the Earth's surface. They have to apply horizontal forces as well, in order to counter the Earth's rotation and get the satellite into the correct orbit.